Question

Two masses 10gm and 40gm are moving with kinetic energies in the ratio 9:25.
The ratio of their linear momenta is
1) 5:6
2) 3:10
3) 6:5
4) 10:3​

Answer 1

For first body

M=m₁

K=k₁

P=p₁

And for 2nd body

M= m₂

K= k₂

P= p₂

Relation between kinetic energy and momentum

K=p²/2m

So, k₁= p₁²/2m₁ and k₂=p₂²/2m₂

k₁/k₂ = p₁²/2m₁ / p₂²/2m₂

2 gets cancel

9/25 = p₁²/10 / p₂²/40

after cancellation we get

9/25 = p₁² * 4 /p₂²

4 gets multiplied with 25

9/100 = p₁²/p₂²

So, p₁/p₂ = 3/10

Answer 2

AnswEr :

Given that two masses 10gm and 40 gm are moving with kinetic energy in the ratio of 9 : 25.

We have to find ratio of their linear momentums. 1)5 : 6. 2) 3 : 10. 3) 6 : 5. 4) 10 : 3.

Now we will use the formula for kinetic energy :-

Kinetic energy (K.E.) = ½ mv²

∴ Ratio of their kinetic energies :-

⇒ 9 : 25 = (½ × 10 × v₁²) : (½ × 40 × v₂²)

⇒ 9/25 = 5v₁²/20v₂²

⇒ 9/25 = v₁²/4v₂²

Squaring root on both sides :-

⇒ √(9/25) = √(v₁²/4v₂²)

⇒ 3/5 = v₁ /2v₂

⇒ 5v₁ = 6v₂

⇒ v₁/v₂ = 6/5

So we can say that,

∴ v₁ = 6 m/s

∴ v₂ = 5 m/s

Now we know that,

Linear momentum = Mass × Velocity [P = mv]

For the first body :

⇒ Linear momentum (p₁) = 10 × 6

⇒ Linear momentum (p₁) = 60 units.

For the second body :

⇒ Linear momentum (p₂) = 40 × 5

⇒ Linear momentum (p₂) = 200 units.

Now ratio of linear momentums :

⇒ Ratio = p₁/p₂

⇒ Ratio = 60/200

Dividing both sides by 20 :

⇒ Ratio = 3/10

⇒ Ratio = 3 : 10

∴ Ratio of their linear momentums = 3 : 10 ⇒ 2) 3 : 10 [Answer]

∴ Correct option : 2)3 : 10 ✔